Well, the topic may be about comparing the speed and responsiveness of Kindle and Kobo, but let's just go ahead and talk about patching software and installing older firmware releases instead.
I use both a Kindle Oasis 2 (reading mobi, azw3, and pdf) and Kobo Aura One (reading epubs) every day. I think the Kindle is generally faster and more responsive. Pages turning, and things happening generally, happen quicker. Words are selected and buttons are registered more accurately. Clicking footnotes and links works quickly and reliably, without having to stick to a specific file format. Text-based PDF files can be read conveniently and reliably using only the page-turn buttons (when in landscape mode) without worrying about zooming to a specific part of the page.
The Aura One is slower. But it is not slower for nothing. It takes longer to turn a page, because it has more words to load, and because it arranges those words more beautifully and elegantly than Kindles do. It hyphenates words, (unlike Kindles), and it hyphenates them well (unlike Nooks). As such, it uses kerning (using spaces between words to spread the words across a line) much more sparingly and elegantly than Kindles do.
If you really want, I could do a video comparison of the two readers. The difference in speed between the devices will be small - they will both be significantly slower than a tablet is. Such a video comparison would not inform you whether you will find the relative speed difference annoying whilst reading. I think the Kobo Aura One is noticeably slower in regular operation, but I think it more than compensates for this by being generally better at displaying text, so I actually find it more pleasurable to read on than the Oasis 2 (except when reading pdf, which I don't even bother to load onto any Kobo device).